I'm studying the rolling motion, but, Why the torque by static friction does work? If the point of application is at rest relative to the inclined plane, therefore, the point of application doesn't move.
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1I’d say all of the work is being done by gravity. Part goes into translation and part rotation, but the force that does all the work is gravity – Ben51 Feb 20 '21 at 01:26
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2Does this answer your question? How can a rigid body's weight do work on it to make it rotate? – Dale Feb 20 '21 at 02:22
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The work done by friction depends on whether the body is rolling without slipping, or slipping. For rolling without slipping the net work done by friction is zero.
For your problem, the object rolls without slipping and only gravity does work. The work done by friction consists of two parts: work for translational motion of the center of mass (negative) and work for rotation motion about the center of mass (positive). The net work done by friction is the sum of these two terms and is zero for pure rolling with no slipping. Your problem shows the work done by gravity and friction for translation of the center of mass as: $mgh-F_rx$; $mgh$ is the work by gravity and $-F_rx$ is the work by friction. Your problem shows the work done by friction for rotation about the center of mass as: $F_r R\phi = F_rx$; this work is due to the torque from the force of friction (gravity has no torque about the center of mass). The total (net) work is the sum of the work for translation plus the work for rotation and is $W = (mgh-F_rx) + (F_rx) =mgh$ as your problem states; note that the net work by friction is zero because the two terms for the work by friction involving translation and rotation sum to zero.
For slipping, friction does net work.
See Consistent Approach for Calculating Work By Friction for Rigid Body in Planar Motion for a detailed discussion of the work done by friction for pure rolling and for slipping.
John Darby
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Without friction the wheel would not roll down but rather slide down. Friction acts to ‘catch’ the wheel in the tangent direction shown in your diagram.
From the wheel’s perspective, something is making it rotate which means Work is being done on the wheel.
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“Work is being done on the wheel” Yes, but you should be clear that the work is done by gravity, not the friction force. https://physics.stackexchange.com/q/614317/ – Dale Feb 20 '21 at 04:17
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@Dale Hi Dale. Came across this post and your comment that the work is done by gravity, not the friction force which made me think about why this situation differs from the role of static friction on a drive wheel to accelerate a car. As I understand it, when a car accelerates without slipping on a horizontal surface the static friction force on the drive wheel responsible for accelerating the car and thus doing work on the car. – Bob D Apr 13 '23 at 14:00
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True the energy for doing that work ultimately comes from the drive train just like the energy in this example comes from gravity. So why is static friction not doing work here while it does in accelerating the car? I would appreciate your thoughts. – Bob D Apr 13 '23 at 14:00
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@BobD no problem! The situation does not differ. There is no work done by the static friction force on the car either. The static friction force transfers momentum, but it does not transfer energy. Momentum and energy are separate conserved quantities. The static friction force is between the earth and the car. It transfers momentum from the earth to the car, so the impulse is non-zero. It does not transfer energy from the earth to the car, so the work is zero. The energy comes from the engine, so the increased KE is not due to any external force doing work on the car. – Dale Apr 13 '23 at 14:21
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Thanks Dale. What bothers me is, for the drive wheel case, the static friction force is the only external force acting on the car in the horizontal direction. Also on many sites it is said the static friction does work. Seems like a lot of confusion out there – Bob D Apr 13 '23 at 15:05
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See https://scripts.mit.edu/~srayyan/PERwiki/index.php?title=Static_friction#:~:text=Static%20Friction%20can%20Perform%20External,static%20friction%20can%20do%20work. – Bob D Apr 13 '23 at 15:19
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@BobD Indeed, there is a lot of unnecessary confusion on this topic. It is very clear if you look at mechanical power: $P=\vec F \cdot \vec v$ where $\vec v$ is the velocity of the material at the point of application of the force $\vec F$. Static friction, like any mechanical force, does work whenever $P\ne 0$. E.g. a box in the bed of an accelerating truck. You don’t have to invent different rules or cases for friction. Any mechanical force follows the same rule in any situation you can analyze with Newtonian mechanics. The link seems fine – Dale Apr 13 '23 at 16:40
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@Dale On last thing. I agree with you that, given the definition of work, the static friction force that accelerates the car should technically do no work since the point of contact with the ground is not displaced by the force. But it causes the car to accelerate, meaning it increases its kinetic energy. Given that, how do we square the work energy theorem with this example? That theorem does not specify the details of the interaction of the force with the object. – Bob D Apr 14 '23 at 15:35
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@BobD the work energy theorem is the root of most of the confusion on this topic. That is more than can be addressed in comments. For the rest, you need to recognize that energy and momentum are separate conserved quantities. It is a mistake to assume that a force transfers energy just because it transfers momentum (or vice versa). The friction force is an interaction between the earth and the car. Momentum goes from the earth to the car, energy does not. It is not a technicality that the force does no work. It is necessary since energy does not transfer from the earth to the car – Dale Apr 14 '23 at 17:52
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@Dale I agree with you if we consider the earth plus the car as the system. Then the torque force and static friction force are an internal action reaction pair and we have an exchange of momentum between the earth and car. What I'm asking about is if the car alone is the system. – Bob D Apr 14 '23 at 18:14
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@BobD I was talking about the car alone as the system. Although my comments wouldn’t change either way. Maybe you should ask a question. We are getting the comments message. I can do a more thorough treatment in an actual answer – Dale Apr 14 '23 at 20:18
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1@Dale But if you were talking about the car alone as the system, why did you say "The friction force is an interaction between the earth and the car. Momentum goes from the earth to the car, energy does not". Any way you're right we're getting comments messages. So let's leave it at that for the time being. Thanks for taking the time to try and help me. – Bob D Apr 14 '23 at 20:59
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@Dale In view of the above, do you agree with John Darby's answer where he says friction does negative translational work and positive rotational work that sum to zero? – Bob D Apr 15 '23 at 11:00
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@Dale OK Dale. You’ve convinced me. Static friction does no work in connection with pure rolling without slipping. Can we therefore, in this case, think of static friction as the mechanism that enables or facilitates conversions between translational motion and rotational motion? – Bob D Apr 15 '23 at 12:35
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@Dale Do you disagree with the answers by John Rennie, John, and Farcher here?:https://physics.stackexchange.com/questions/346660/work-done-by-static-friction-on-a-car#:~:text=In%20accelerating%20the%20car%20from,by%20static%20friction%20is%200. John Rennie has an interesting take on this in his comments that has me thinking further – Bob D Apr 29 '23 at 13:42
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@BobD the idea of John of positive linear work and negative rotational work is probably not wrong. But it is rather ad hoc for my tastes. I prefer simple rules that can be universally and consistently applied. I think Farcher is wrong to say that work is done when a force undergoes displacement. A force has no mass, so its displacement is immaterial. It is the displacement of matter that defines mechanical work. I think John Rennie is simply wrong, external work is not done. If external work were done then the internal energy would not decrease. There is a lot of confusion on this topic – Dale Apr 29 '23 at 14:25
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@Dale What’s wrong about thinking of it this way: Move the static friction force from the point of contact to the COM and add a counter clockwise couple to account for its torque about the COM. The counter clockwise couple reduces the rotational KE of the otherwise free spinning drive wheel, and causes acceleration of the COM giving it translational KE. Wouldn’t the static friction force now be doing work? – Bob D Apr 29 '23 at 15:10
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@BobD first issue is a global conservation of energy issue: if the static friction does work then why is the car losing internal energy? Second is a local conservation issue: how does the mechanical energy get to the CoM? The global one can probably be ad hoc justified, but I would like a rule that can be consistently applied. The local one is more challenging. The product of the stress tensor and the velocity gives a mechanical power density vector. That describes locally how power flows through a machine. It goes to 0 at the contact patch. So there is no power flux from road to CoM – Dale Apr 29 '23 at 16:59